Learn Hardcore Finnish: Laitan muistilapun pöydälle, jotta muistan soittaa siskolleni huomenna. - I put a sticky note on the table so that I remember to call my sister tomorrow.

Because it’s the direct object of laitan (I put). With many common verbs like laittaa, Finnish typically marks a completed/whole object with the -n form (often called the accusative; it looks like the genitive in the singular).
So muistilappu → muistilapun = (the) note as a whole item you’re putting somewhere.

What case is pöydälle, and what does it mean exactly?

pöydälle is the allative case (-lle), meaning onto / to the surface of something.

pöydälle = onto the table
Compare:
pöydässä (inessive -ssa) = in the table (inside it)
pöydällä (adessive -lla) = on the table (location)
pöydältä (ablative -lta) = off/from the table (surface)

Why is it Laitan (present tense) if the action could be in the future?

Finnish often uses the present tense for near-future or planned actions. Context supplies the timing. Here, even if you mean I’ll put a note (now / soon), laitan is natural.

Why does Finnish use jotta here, and what does it require?

jotta introduces a purpose clause: so that / in order that.
After jotta, Finnish uses a normal finite verb form (here muistan, present). So:

jotta muistan… = so that I remember…

Could this be että instead of jotta?

Not with the same meaning.

jotta = purpose (so that / in order to)
että = content/result (that)
So jotta muistan = I’m doing this for the purpose of remembering, which fits the sentence.

Why is it muistan soittaa and not something like “to remember that I will call”?

Finnish commonly uses muistaa + infinitive to mean remember to do something:

muistan soittaa = I remember to call
If you wanted remember that… (a fact/memory), you’d more likely use muistan että…:
muistan, että soitin = I remember that I called

What form is soittaa here?

It’s the first infinitive (dictionary form). With muistaa, the infinitive expresses the action you need to remember: remember to call.

Why is it siskolleni and not siskoni or minun siskolle?

siskolleni breaks down as:

sisko (sister)
-lle (allative: to) → siskolle = to (a) sister
-ni (possessive suffix: my) → siskolleni = to my sister

You can say minun siskolleni, but it’s often unnecessary because -ni already marks possession.

Does -ni always mean “my”? Is it required?

Here -ni means my, and it’s very common in this kind of phrase. It’s not always strictly required in casual speech (people sometimes use just siskolle if the context is obvious), but siskolleni is clear and standard.

Why are there no explicit pronouns like “I” in the sentence?

Finnish verb endings show the person, so subject pronouns are often omitted:

laitan already means I put
muistan already means I remember
You can add minä, but it usually adds emphasis/contrast.

Why does muistilappu change to muistilapun (the pp → p change)?

That’s consonant gradation. When certain endings are added, a “strong” consonant can become “weak”:

muistilappu (strong pp) → muistilapun (weak p)
This happens in many Finnish words depending on the grammatical form.

Could the word order be different?

Yes, Finnish word order is flexible. This version is neutral and natural. You could move parts for emphasis, e.g.:

Huomenna laitan muistilapun pöydälle, jotta muistan soittaa siskolleni.
That emphasizes tomorrow. The cases/endings keep the roles clear even if the order changes.